boundedness of solutions of duffing’s equation* · journal of differential equations 61, 178-207...

JOURNAL OF DIFFERENTIAL EQUATIONS 61, 178-207 (1986)

Boundedness of Solutions of Duffing’s Equation*

TONGREN DING +

School of Mathematics, Institute for Mathematics and its Applications, University of Minnesota, Minneapolis, Minnesota 554.55

Received June 21, 1984; revised November 30, 1984

J. Littlewood, L. Markus, and J. Moser proposed independently the boundedness problem for solutions of Dufting’s equation: a+g(x)=p(t), where p(t) is continuous and periodic and g(x) is superlinear at infinity. The purpose of this paper is to prove that all solutions of the above-mentioned Dufing’s equation are bounded for t E IR when p(t) is even (or when p(t) is odd and g(x) is odd). 0 1986 Academic

Press, Inc.

I. INTRODUCTION

The study of the bounded solutions of Hamiltonian systems is closely related to the problem for the stability of motions, for example, the problem of collisions for n bodies [ 11. It challenges one’s attention since estimating the bounds for solutions of Hamiltonian systems is really not easy.

J. Littlewood [3], L. Markus [2], and J. Moser [l] proposed independently the boundedness problem for solutions of Duffing’s equation

a+gb)=p(t), (1)

where p(t) is continuous and 2rc-periodic in t E R, and g(x) is continuously differentiable in x E IF! and satisfies

lim $ g(x) = co as Ix/ -9 00.

Up to now, the only contribution to the above problem is the work of G. Morris [S], in which he proves that each solution of the special Duffing’s equation: 2 + 2x3 = p( t), is bounded for t E R. The purpose of this paper is to prove the following theorems.

* This research was supported in part by NSF Grant MCS 79-01998. + Present address: Department of Mathematics, Peking University, Beijing, China

178 0022-0396186 $3.00 Copyright 0 1986 by Academic Press, Inc. All rights of reproduction m any form reserved.

DUFFING'S EQUATION 179

THEOREM 1. All solutions of the Duffing equation (1) are bounded for t E R! whenever p(t) is even.

THEOREM 2. All solutions of the Duffing equation (1) are bounded for t E R whenever p(t) is odd and g(x) is odd.

Our method is based on a careful study of some Lagrange stable sets together with the twist property of the Poincare map. The method of stable sets developed in this paper can be generalized further to measure-preserving homeomorphisms in n-dimensional space, and is useful in studying the bounded solutions for Hamiltonian systems [4].

II. PRELIMINARIES

In what follows, we will need some basic facts stated in the following propositions.

PROPOSITION 1. Let {A, > be a sequence of nonempty connected compact sets in R2, and let

Then the intersection flF=, Ak is nonempty, connected and compact (see C6, P. 1631).

Let CD: R2 + R2 be a homeomorphism, and let G c R2 be a bounded connected open set. Then the image Q(G) is also a bounded connected open set in R2, and &D(G) = @(aG), where 8 is the boundary operator for sets in R2.

Let z0 E R2 be a fixed point of CD. For any r > 0, denote by B, the open disk in R2 centered at the origin 0 with radius r, and denote the boundary aB, by S,. Let G c R2 be a connected open set with z0 E G. Assume r > Iz,, /, where JzO ( is the distance from 0 to zO. Then Q(G) n B, is a nonempty open set with z0 as an interior point. Set

G; = a,,C@(G) n 41,

where a,,[.] denotes the largest connected subset of [.I containing z0

PROPOSITION 2. Let G c R2 be a connected open set, and let zO E G be a fixed point of @. Let r > Iz,, I and assume that

180 TONGREN DING

where G denotes the closure of G. Then we have

Proof: First, we assume that

Q(G) c B,.

Then we have

(3)

G) = aJ@(G)n B,] = oJ@(G)] = Q(G).

It follows from (3) that Q(G) n S, = @. Then (2) yields the desired conclusion

aGj n S, = a@(G) n S, = @( dG) n S, # 0.

Next, assume (3) is not true. Then, since Q(G) is open, there is at least one point p E G such that

Q(P) E R2\R

Since z0 and Q(p) belong to the connected open set Q(G), there is a continuous curve

r: z = z(t) (0 < t < 1):

in Q(G), with z(0) = z0 and z( 1) = Q(p). Hence, there is a unique t, E (0, 1) such that

z(t) E @j(G) n B, for f E [0, to),

and z(t,) E Q(G) n S,. This means that

z(t,) E ac; n s,,

and thus 8Gb n S, # 0. The proof of Proposition 2 is thus completed.

Let x1 < x2 and define

and

W={(x,y)ER21X1<X<X2, -co<y<co},

L(x/J= {(x,y)ER*IX=Xk, --co <y< co} (k = 1, 2).

PROPOSITION 3. If E c R’* is a bounded, connected open set and satisfies

EnL(xk)ZO ,for k=l,2,


then there is a connected closed set A in aEn m with

AnL(x,)#@ fork=l,2.

(We will prove this proposition in the Appendix.)

PROPOSITION 4. Let

r: z = z(t) (0 d t 6 1 ),

be a continuous curve in W with z(0) E L(x,) and z( 1) E L(xz). Let Fc W be a connected set. If there are two points p, and pz of F such that p, is located above F and p, is located below F, then we have

(The proof is easy.)

I-nF#@.

III. LAGRANGE STABLE SETS

A connected closed set Kc R* is said to be simply connected if for any bounded open set E, “aE c K” implies “E c K’.

Let @: R* + R* be a homeomorphism. A set Fc R* is said to be positively (or negatively) Lagrange stable for @ if, for any point p E F, the sequence

{@(p):k= 1, 2 ,..., (ark= -1, -2 ,... )}

is bounded.

LEMMA 1. Zf @: R* + R ’ is an area-preserving homeomorphism with a fixed point z,,, then .for any constant r > jzO /, d> has a positively Lagrange stable set K,+ and a negatively Lagrange stable set K, satisfying

(i) K,+ and K, are simply connected closed sets passing through z,; (ii) K,f nS,#@, K; nS,#/ZI, K,+ UK; cB,; (iii) @(KT) c K,!, W ‘(K;-) c KY-.

Proof: We will break the proof up into the following steps. Step 1. Let r > IzO 1, and set Hf. = B,. Then we have

z. E H; n @(HJ); RfnS,#@.

If @(Jjf ) n S, = 0, then we have

182 TONGREN DING

It follows that H;\@(Ri) is a nonempty set. Then we have

mes[HF] - mes[@(Rf)] = mes[H;\@(Rj)] > 0,

which is in conflict with the fact that @ is an area-preserving homeomorphism and mes [aHi] = 0. Hence, we have

Then set

HP = oz,[@(H~) n B,].

It follows from Proposition 2 that

Rp-b!?,#525, (mes[aHf] = 0).

Since HF c @(Hf. ) n B, c B, = H: , we have

@(Hf)c@(Hf). (4)

Note that @(Hz) is a bounded connected open set with z0 E @( HF). Assume @(RF) n S, = a. Then we have

@(Z?f) c B,.

It follows from (4) that

@(HS) = @(Hi) n B,,

and consequently

~(H~)ca,,[~(H~)nBB,]=HS.

Hence, the assumption @(A:) n S, = @ implies that H;\@(Rf) is a nonempty set, and thus

mes[Hf] - mes[@(gF)] = mes[H:\@(Rf)] > 0,

which is in conflict with

mes[Hz] = mes[@(Hf)] = mes[@(flf)].

Then we have

@(RF) n S, # a.

Now, set Hz = o,[@(H~) n B,].

DUFFING’S EQUATION 183

It follows from Proposition 2 that

AjnS,#0, (mes[aH,3] = 0).

Since Hz c @(H:) n B, c @(H,‘) n B,, we have

H,3 c a,[@(H;) n B,] = H;

and

@(Hj) c @(H?).

Note that @(H,3) is a bounded connected open set in R2 with z0 E @(H:). Using the same technique as mentioned above, we can prove by induc-

tion that

Hf+ l= a,[@(Hr) n B,] (k = 1, 2,...),

are connected bounded open sets with

(5)

z,EH~ and &?nS,#@ (k= 1,2,...).

Step 2. Set

K =,fi, R; (~>IzoO.

It follows from Proposition 1 that K,- is a connected compact set. Moreover, we have

zonk;, K; c & K; n S, # 0.

Since Hf + ’ c @(H$), we have

@-yR~+‘)cA~.

Then, from (5), we get

K,= fi j+ fi R;+‘cjj,. &=I k=l

Hence,

@-‘(K;)= fi @-‘(R;+‘)c fi R;=K,-, k=l &=I

which also implies that K,- is negatively Lagrange stable for @.

184 TONGRENDING

To indicate the related homeomorphism @ and the fixed point zO, we denote K; by K;(z,, @).

Step 3. Since @-I is also an area-preserving homeomorphism with fixed point zO, we get a connected compact set K; (z,, @- ‘). Define

K: = K,?(z,, @)= K,(z,, @-I).

Then K,+ is a connected compact set with

zo E K: > K+ nS,f12(,

It can be seen by definition that

K,+ c B,.

which also implies that K: is positively Lagrange stable for @.

Step 4. It is not hard to verify that all al: (k = 1, 2,...) are simply connected sets. Hence, K; is a simply connected closed set, and so is K,? .

The proof of Lemma 1 is complete.

With an observation on the proof of Lemma 1, we can easily verify the following conclusion.

LEMMA 2. For any constants r and s with s > r, we have

K: c K,t , K; c K; .

IV. TWIST PROPERTY OF THE POINCARB MAP

The Dufhng equation (1) is equivalent to a Hamiltonian system

f = qt, 4 Y), I; = -fcc(t, x, Yh (6)

where

H(t, x, y) = jy’ + sx g(x) dx-p(t) x for (t, x, y) E R3. 0

Let (u, v) E IR’. Denote by

x = x(t, u, v), Y =y(t, 4 VI,

the solution of (6) satisfying the initial condition

x(0, 24, 0) = 24, y(0, 24, v) = v.

(7)


It is known that the solution (7) exists for ail t E R and is differentiable in (t, U, v) E R3. Moreover, the Poincare map

@: FP + LIP; (u, u)b-+ (x(271, u, u), y(27c, 24, II)),

is an area-preserving diffeomorphism. It can be proved that the Poincare map @ above admits an infinite number of fixed points zi (i = 0, 1, 2,...) (see r-71).

Let u = Y cos 8, u = Y sin 9 (r > 0), and let

x=Rcos@, y = R sin 0,

where R = R(t, r, e) = [x*(1, u, u) +y*(t, u, u)]“* is continuous in (t, r, 0) E R x [0, cc) x R, and 2x-periodic in 8. It can be seen that there is a constant a,, > 0 such that

R(t, I, e) > 0 for t E [0,27~] and 6 E R,

whenever r > a,. (In fact, consider the solid cylinder

Zb={(t,X,y)~Odt<27c,06X2+y2~b2},

where b is a positive constant. Denote by

x = x(t; 7, z4, u), y = y(t; 7, 4 u), (0 6 t G 2n)

the solution of (6) satisfying the initial condition

x(7; 7, u, u) = u, y(7; 7, 2.4, u) = u

with the initial point

Then all the integral curves defined above constitute a compact tube T. It follows that

TcZ,

for some constant d> 1. Let a, > d, and let

u* + u* > ai (i.e., (0, u, 0) # Z,,).

Then every integral curve

x=x(t,u,u)=x(t;O,u,u), y=y(t,u,u)=y(t;O,u,u) (O<t<27r)

186 TONGREN DING

cannot intersect the above-defined tube T. Since T contains Z,, we have

R( t, Y, e) = Jx’( t, u, u) + y’( t, u, u) > 1 (O<t<271 and 8~lR)

provided that r = dm > a,, which is the required conclusion.) In this case, (6) is equivalent to the following system

d = [R cos 0 -g(R cos 0) +p(t)] sin 0,

6 =A g(R cos 0) cos 0 - sin*@ + Ap(l) cos 0.

It follows that R = R(t, r, 0) ( >0) and 0 = O( t, r, 0) are continuous in (t, r, 0) E [0,27c] x [a,, co) x R, where R(0, r, 0) = r and O(0, r, 0) = 8. Moreover,

O( t, r, 8 + 27r) = O( t, r, 0) + 271.

Hence, for r > a,, the Poincare map Q, can be put into the following form:

P = R(271, r, Q), f$ = 0 + h(r, 0), (8)

where R(27c, r, 0) and h(r, ~9) = 0(2x, r, 0) - 8 are continuous in (r, 0) E [a,, co) x R and 2z-periodic in 6’ E R.

The following lemma is just Lemma 4.3 in [7], which expresses the twist property of the Poincare map @ for the Dufling equation (1).

LEMMA 3. For any given integer m > 0, there is a constant a,( > uo) such that

h(r, 0) < -2mz for r>u, und 0E(W.

V. PROPERTY A

@ for the DufIing Let z,, and zb be fixed points of the Poincare map equation (1 ), and let

K: = K,+(%, @), K, = K; (zb, Cp 1

for r> lzOl + lzbl (see Lemma 1).

DEFINITION. The set K,+ (or K,) (for r > IzO) + lzbl) is said to possess Property A if and only if for any constant d> lz,, 1 (or d> Izb I ) there is a constant s > d and a fixed point zd of @ such that

zd E K,‘\B, (or, zd E KS-\B,).


Now, we prove

LEMMA 4. If K,? and K; possess Property A, then all solutions of (1) are bounded for t E R.

Proof: First, we claim that for any constant a> 0 there is a constant b > a with

B,cK,+. (9)

Assume the contrary. Then there is a constant r0 > IzO 1 such that

(Br\K,+ 1 n B, f !3 forr>r,. (10)

Without loss of generality, assume r,, > a,, > IzO 1, where a, is the constant chosen in Section IV. Now, choose constants rl and r2 (rz > rl > rO) with

B, = @l&y I= 4,. (11)

Since K,’ has Property A, we can take r2 such that @ has a fixed point z, with

z, EK: nsr, for some r3 > r2.

Then there is a connected closed set F, c K: with the property that

and F, connects zJ to S,,, where R[r,, r3] is the closed annulus bounded by S,, and S,,. Let

ho = maxIh(r, O)l for r E [r,, rJ and 0 E R,

where h(r, 0) is defined in (8). Then take two integers 1 and m satisfying

I> h,/2n, m>l+ 1. (12)

Using Lemma 3, we choose a constant do > r3 such that

h(r, 0) < -2mrc forr>d, and OER. (13)

Similarly, choose constants d, , d2 (d2 > dJ > d, > r3) satisfying

B, = @(&I, I= B, 9 (14)

such that there is a fixed point

188 TONGREN DING

Then we can take a connected closed set F2 c K$ in R[r, , d,] containing z2 and connecting z2 to S,,.

From (lo), we have

where w is a positive integer or infinity, and each Ei c B, is a nonempty connected open set with

aEi c K; v S, (i = l,..., W).

Since K; is a simply connected compact set with Kz n S, # @, Ai=(aEinS,,)\K$ is an open arc of S, (i= l,..., w). It follows from (10) that there is at least one set, say E, for definiteness, such that

-4nS,Z0, (A, Z0).

Now, we are going to prove the following inequality:

K,:nE,#0. (15)

To this aim, take a continuous and simple curve

C:z=z(t)=(r](t)coscc(t), r(t)sincr(t)) (0 < t < 1 ),

in (E, u A,)n R[r,, d2] with z(O)ES, and ~(1)~s~~. Since ~J=cY(z) is determined up to 27r, we have

8 = a(t) = ccj( t) = ao( t) + 2jrc (jG z),

where cl,(t) is a single-valued continuous function (0 Q t < 1) with a,(O) = 8,, E [0,2n) and c1J 1) = 8, uniquely determined.

Now, consider (r, 0) as the Cartesian coordinates of the (r, 0)-plane. Set

T,={(r,@ r>aand8ER}.

Let @*: T,, + T,, be a mapping defined by (8); i.e., @* is a lifting of @ in R2\BII,.

Then we have an infinite number of continuous curves

c(‘): Y = u](r), e=cc,(l) (O<t< l),

in T,(i E Z), which are liftings of C. Since C n K$ = 0, and Kz connects S, and S,, then

c(i) n c”’ = 0 whenever i #j.


Denote by L, the vertical line r = a in TO; i.e., L, is the lifting of S,. Then Cci’, C(i+l), L,,, and L, bound a simply connected open set Vci) in TO for iEZ.

Note that F2 c K+ and F, c K+ c K+. Let z ’ zy), FIi’, and Fy) be the liftings of zl, z2, F d2 and F,, re$ective?y, with ‘I”’ 1,

,(li) E jji) c V(r) 2 zy’ E F$f’ c v(j) (iEZ).

There are two cases to be considered; i.e.,

Case I. F, n F2 # 0,

Case II. F1 n F2 = 0.

For Case I, set

F= F, uF~.

Then Fc R[r, , d, ] n Kz is a connected closed set containing two fixed points z1 and z2. Let

pi) = F$‘) u F(i) 2 2

Then F(” c Vci) (i E Z) are liftings of F. Because of (11) and (14), the connected closed set @*(PO’) is located

between L, and L,. Since z, and z2 are fixed points of @ with z$O), z$O) E F(O), we have

r@*(p) = z(k) 1 9

@*(zy’) = zp

for some integers k and II. Hence,

@*(PO’) n Pk’ # 0, @*(FO’) n P) # 0.

Since zr is a fixed point of @ satisfying

we have

and

z(lO) = (r2, eyy E P’ c v(O)

zp = (r2, f$j”‘) E I+’ c V(k),

where d\“‘= 19)“’ + 2kx. From (8), we have

f$$O’ = 8’,0’ + h(r,, 8’,0’).

190

It follows that

TONGREN DING

k = h(r,, @i”‘, -ho 2n

a---> -1. 271

In like manner, we get

iZ= Mdo, q”‘, 27c

< --m

provided that do is sufficiently large. Hence, we have

k> -I> -m+13n+l.

which implies that Vck) n I/‘“’ = @. It follows that z\“) is above I?“‘) and zp) is below C’-“‘. Using Proposition 4, we have

@*(PO’) n Cc-“’ # 0,

which yields

@(F)nC#0. Note that

Q(F) = @‘(Kd:) = K;, CcE,uA,, @(F)nA,=@.

Then we conclude that (15) holds in Case I. Now, we turn to consider Case II. Let W be the open strip bounded by

L,,- and Ld; in To, where G!‘, is some constant (d,<d’, dd,) such that F$O) c @ connects L,, and Ld; . Then we have two simply connected open sets W, and W, in W separated by F(p) such that W, extends to infinity in the positive direction of the 8 axis and W, extends to infinity in the negative direction of the 0 axis.

Since F, n F2 = $3, we have

Fy) n F$j) = 0 for any i and j.

It follows that Fi-‘) is below F$O); i.e.,

F(l-“c W,v L,,.

Note that z(,-‘)E F’,-“n W,. Now, we have

P(z {-l))=Z(lk-I), @*(q’) = zp’

for some integers k and n. By (12) and (13), we have

k>,-I>-m+l>n+l. (16)


Note that @*( W,) is a simply connected open set extending to infinity in the negative direction of the 8 axis, and its boundary a@*( W,) satisfies the relation

a@*( W,) c @*(F$oJ) u @*(L,,) u @*(L,; ).

Without loss of generality, we assume that B,, c @(B,). It follows from (11) that

@*(Fy’) n L,, # 0.

Moreover, @*(L,,) is at the left of L,, and @*(L;) is at the right of L,?. Hence, there is at least one point q E @*(F$O)) A L,,, which is above z\“- ‘I, (Zl ck ~ ‘) E Vck- ‘) n L,, n W,). It follows that q E @*(F$O)) is above Cck ~ I). Since @*(z$O)) = zy)E vcn), zp)E @*(F$O)) is below C’” + ‘I. Because of (16) Cck-‘) is above C(“+‘) or C(k-l)=C(n+l’. Then @*(Fzo)) has a point q above C?+‘) and a point z2 cn) below C(“+ I). Using Proposition 4, we have

cD*(F$“‘) n ccn+ l’ # 0,

which yields

@(F,)n C# 0.

It follows that (15) is also valid in Case II. However, (15) is in conflict with the definition of E, . This contradiction

proves the desired conclusion, (9).

In a similar way, it can be proved that for any constant a > Izb 1 there is a constant b > a such that

B,cK;. (17)

By (9) and (17) for any constant a > 0, there is a constant d > a such that B, c Kd+ n K; It follows that B, is a (positively and negatively) Lagrange stable set for the Poincare map @ of (1). Then, for any initial point (u, u) E B,, the solution (7) is bounded for t E R. Since a is arbitrary, Lemma 4 is thus proved.

VI. PROPERTY A:

Now, assume that K,? (or K;) does not possess Property A. Then there is a constant 6, > 0 such that K,+\B,, (or K; \Bho) contains no fixed point of @ for any r > b,. Let

B,\K,T = [ E,:, i= 1

or, B,\K; = [ Erz , i= 1 >

192 TONGREN DING

where Er> (or E,zz) is a simply connected open set (i= 1, 2,...). Since @ has infinitely many fixed points {zj} with lzjl -+ CC as j + 00, we have

Zk E E,t,, (Or Zk E Eryikl forr> Izk( >6,.

Then we have to consider the following properties.

PROPERTY A,*. There is a constant b, > b, such that for any zk E Ecik (or zk E E,,,,) with b, < (zk/ <Y, we have

PROPERTY A:. For any constant h > b, there is a fixed point zk E E,t,, (or zkE E,.) such that

a-q,, n Sh = 0 (or iYE,, n Sh = 0)

with Y > Jzk ) > b.

It follows that if K,? does not possess Property A, then it possesses Property A: or Property A:. Similarly, if K; does not possess Property A, then it possesses Property A: or Property A:.

LEMMA 5. K,+ and K, do not have Property A:.

Proof Since the proof is very similar to that of Lemma 4, we only give some key points and omit the details for the proof below.

Assume that K,? has Property A:. Then, choose constants rO, rl, rz, d,,, d,, and d, (b, < rO< r, < r2 <

d, < d, < d,) such that (11) and (14) are satisfied. Moreover, because of Property AT, we can choose two fixed points

zl E Ed:,,, n Sr17 z2~E&p&,~

with

aE&, n s,, Z 0, aE$,, n St,, # 0.

Note that

aE$, i, n s, f 0, aEA,, n s, f 0.

Then there are continuous and simple curves

and

CICE~:,~~~RC~,, 41 joining z, to a point q, ES,,

CZ c Ed:,iz n Nr,, 41 joining z2 to a point q2 E S,, .


Moreover, there is a continuous and simple curve

cc b%, u4,)n~Cro~ 41

connecting S,, and S,, where A ;, = dE2, ;, n S, is a nondegenerate closed arc of S,.

Since E&, is open, we can assume that C, and C, do not intersect C. Using Proposition 3, we get a connected closed set

Fc (a&-& I, nK,:)nRCr,, d,l

connecting S,, and S,,. Denote the liftings of z,, z2, C,, C,, C, and F by zii), zr), Cl’), Cy), I?),

and pi’, respectively (in Z). Let Vti) be the open set bounded by Cc”, C(‘+‘), L,, and L dz. Note that

PO) n Cli) = @,

and Cl” joins zj’) to q\” E V”‘n L,, (in Z). It follows that zj~- I) is below F”’ (i.e., z\-” belongs to W,) (see the proof of Lemma 4).

Similarly, it can be seen that zi’) is above F”’ (i.e., z$l) belongs to I%‘,). Then, employing a similar method used in the proof of Lemma 4 for

Case II, we can prove

@*(F(O)) n Cc”+ I’ # a.

It follows that

which is a contradiction. This proves that K,? does not possess Property A I*.

In a similar way, it can be proved that K, does not possess Property A:. Lemma 5 is then proved.

VII. MUTUAL SYMMETRY OF K,t AND K;

Now, assume that the 2rc-periodic function p(t) is even; i.e., p( - t) =p( t), for t E R.

Let

5=x, fl= -y, T= -t. (18)

Then, from (6) we get

5’ = v, ‘I’= -s(5) +p(th (19)

194 TONGREN DING

where 5’ and q’ are derivatives of 5 and q with respect to t. It follows from (7) and (18) that

t_=x(--z, 4 u), rl= -.?J(-&U,V) (ZE[W),

is a solution of (19) satisfying the initial condition

5(O) = u, r](O)= -v. (20)

On the other hand, since (6) and (19) are equations in the same form, we conclude from (7) that

is a solution of (19) satisfying the initial condition (20). Then the uni- queness theorem yields that

x(5, u, -v)=x(-T, u, v), Y(? u, -0) = -Y( -? 4 v)

for z E R. Hence, we have

x(-27c,u, -u)=x(2n,u,v), y(-2n,u, -v)= -y(2n,u,u), (21)

for (u, u) E BP. Note that

@: (u, u)- (xG37, 4 v), Ych u, v)) for (2.4, v) E R2

@-I: (u, -V)H(X(-2x, u, -v),y(-27?, 24, -v)) for (u, --u)E R2.

It follows from (21) that

@-I: (4 -VI + b427h 4 VI, --YP, u* 0)) for(u, v) E R2,

Therefore, @ and @- ’ are mutually symmetric with respect to the x axis in the sense that @(x, y)= (u, v) if and only if W’(x, -y) = (u, -v). It follows that if z0 = (x,, y,) is a fixed point of @, then zb = (x,, -yO) is a fixed point of W1 and thus a fixed point of @J. Hence, the set of fixed points of CD is symmetric to the x-axis in R2.

Using the symmetric property of @ and @ - ’ mentioned above, we can verify directly that K,+(z,, @) and K;(zb, @) (= K,f(z&, Q-l)) are mutually symmetric with respect to the x axis in R2.

In summary, we obtain the following


LEMMA 6. Assume that p(t) is even and zO = (x,, yO) is a fixed point of the PoincarP map @ of (1). Then zb= (x,,, -yO) is a fixed point of @, and K,? = K,?(z,, @) and K; = K;(zb, @) are mutually symmetric with respect to the x axis.

In a similar way, we can prove

LEMMA 7. Assume that p(t) and g(x) are odd functions. Let zO = (x,, y,) be a fixed point of @. Then zb = (-x0, y,,) is a fixed point of CD, and K,! = K,+(z,, Cp) and K; = K; (zb, @) are mutually symmetric with respect to the y axis.

VIII. MINIMUM SETS

From now on, we assume that Lemma 6 or 7 holds. Then @ and @ -’ are mutually symmetric (with respect to the x axis or the y axis). The set of fixed points of @ is then symmetric to the x axis or the y axis.

Let zO and zb be two symmetric fixed points of @, and let K; = K;t(z,, 0) and K,- = K; (zb, @). Then K,? and K,- are mutually symmetric to the x axis or the y axis. Let z1 and z’, be two symmetric fixed points of @. Then z, E B,\K,+ if and only if z; E B,\K, .

Denote the interior set of KT by int[KT 1. Then we have

K,! =int[K,?]uCYK,?.

Since @ is an area-preserving homeomorphism and @(K,? ) c K,? , we get

@(int[K;t])=int[K:], @(cYK,? ) c aKj+ . (22)

Set

Since K;’ is a simply connected closed set in R*, G+ is a simply connected open set in the extended plane (i.e., in R2 u (cc }) and 8G + = 8K:.

Let z, E B,\K,! (r > Iz, I), and let

B,\K,? = 6 E,?. i= 1

Then, without loss of generality, assume that z1 E ET, where ET is a simply connected open set in G+, and A,+ = aE: n S, is a nondegenerate closed arc of S,. Let F: = aE[ n K,+ . Then F: is a simply connected closed set in aK;t , and ET is bounded by F: and A:.

505 hl 2-4

196 TONGREN DING

Assume that r is a sufficiently large constant. Then there are constants r, and r2 (r > r2 > r, > Izi I) such that

k, c @(Br,) = B,> (23)

which implies that @(S,,) n B,, = @. We can choose an open arc C: of S, in ET and a simply connected closed set J,+ c F: such that C+ and J: bound a simply connected open set H+ c ET with z1 E H,+ , for i = 1,2. Then we have

J: cJ$, H:cH;cG+.

LEMMA 8. Let JT and Jj+ he defined as above. Then we have

J: c@(J;)cF,+.

Prooj First, we claim that

@(J:)nJ: #CL

(24)

(25)

Since @ is area-preserving, we have

@( Hzf ) n int [K,? ] = @,

which yields

@(H:)cG+.

Hence, we have

@(i?Q) c CT+, (II: c CT’).

Now, assume that (25) is false. Then @(J: ) n J: = 0. Note that @(Jz ) n H: = @ and @(J: ) n

C: = 0. It follows that

Since z, E @(H:), z, is surrounded by c?@(H:)( = @(J:) u @(C:)) in G’. It follows that

@(CT ) n R, f 0,

which is in conflict with (23). This contradiction proves the desired conclusion, (25 ).

Note that J: c F:, @(J:) c a&+, and @(J:) n G+ = 0. Then, (23) and (25) imply (24). The proof of Lemma 8 is thus completed.


Let

1; = @(JT) for i= 1, 2.

Then we have

I: cl,+ cF: caKr+,

and (24) is equivalent to

@-‘(Z:)cI; cF:.

Similarly, z; E B,\K,- (r > lz’, I), and

B,\K,- = fi E,: . i= I

(26)

Without loss of generality, assume that z; E E; , where E; is a simply connected open set. Let

F;=aE;nK- r 3

and so on. In a similar way mentioned above, we get

and

@(I, )cZ, c F,, (27)

where I;, I, and F; are symmetric to I:, I:, and F,+ (with respect to the x axis or the y axis), respectively.

Let C be a continuous and simple curve in

(B,\K,+)nNro, r,l (or (B,\Kr-)nNro, r,l)

joining S,, and S,,. Define a continuous function

I)(Z) = arg z - 8, for z fz C,

where 8, = arg pO(O < 8, < 27~) and p0 is the end point of C on S,,. Then, with an observation of the method employed in the proof of Lemma 4, one can get the following conclusion without difficulty.

For any given constant CI > 0, we have

sup CWH > c~ or inf [$(z)] < -c(, (27*) ZEC ztc

whenever r, is sufficiently large.

198 TONGREN DING

In fact, assume the contrary. Then there is a constant a,>0 with the property that

-ao<W)<ao (ZE C) (*)

for any ri > Ye. To use the convenient notations as in the proof of Lemma 4, we replace

rl by d2 temporarily, and choose constants ri, r2, d,,, d, (r. < r, < r2 <do < dl < d2 < r) such that

4, = @(B,,) = Br, 3 B,c @(B,,)cB,.

Let C@) (i E Z) be the liftings of C in TO. Then Cci), Cc’+ “, L,, and L, bound a simply connected open set Vci’ in TO for i E Z.

Using Proposition 3 together with the property that K,f is a closed connected set joining S, and S,, we get a closed connected set /icaK: nR[r,, d,] joining S,, and Sd,. Let Aci’c V(j) (ieZ) be the liftings of ,4 in TO. Note that

AcK,+, @(A)c@(K:)cK:, CnK,? =a.

Then we have

Cn@(A)=IZ(,

which implies

p (-) @*(A’o’) = Qj (iEZ).

Since the connected set @*(A(O)) is located between L,, and L,, we have

CD*(P) c V’“’ for some integer s.

Now consider the strips

Q@‘= {(r, 0)) r. 6 r < d,, 8, -do + 2(k - 1) rr 6 0 6 B. + do + 2(k + 1) rc}.

for k E Z. It follows from (*) that

V(k) c Q’k’ (k E Z).

Then we have

@*(/1(o)) ,= pa’ c Q’G.

Choose two points

P ‘O’=(r,,BjP’)E/l(0)nLL,,cQ(O’


and

q”’ = (d, , Or)) E A (‘) n Ld, c Q(O).

Note that

10(O) I P

and Ify’I < a, = IfI, I + a, + 271.

It follows that

@*(p(O)) = (i,, c,dip') E Q'"', @*(q”‘) = (ci,, 4:“‘) E Q(‘),

where i,, #?I, d,, and 4:) are determined by (8). Then we get

@O’ = e(O) + h(r,, e(O)) P P P ’

q+O) = e(O) + h(d, ) e(O)) Y 4 4

with

and

4) - &I + 2(s - 1) 7.c - c(1 < h(d, , ep, < 8, + a, + 2(s + 1) It+ Lx,.

Hence

Ih(d,) fy, - h(r, ) l3lp’)I < 2(a, + Lx1 + 27c).

Note that h(r, , f?lp)) is bounded for fixed r, Then h(d,, HI$)) is bounded for arbitrarily large d,. However, this conclusion is in confltct with Lemma 3. Hence, the required inequality (27*) is established.

To prove the intersections of Z: and I,-, choose a simply connected closed set

A1 = 1: n Nro, 41

joining S, to S,, and consider its liftings

A;‘) c V(i) (iE Z).

It follows from the above-proved inequality that A, intersects x axis and y axis many times whenever rl is sufficiently large and so does I,+. Then Z: and I; intersect each other, since they are mutually symmetric with respect to x or y axis.

Moreover, there are simply connected closed sets

A+ CZT, A- CZ,

200 TONGREN DING

,Fi

FIGURE 1 FIGURE 2

/ /F;

F,

such that A + and A- bound a simply connected open set D c ET with z1 E: D, provided that rl is large enough. It follows from (24) and (27) that

@(n+)cF:, @(A-)cF, (28)

which are basic conditions for the following discussion. To make the arguments clear, we use the briefly sketched Figs. 1 and 2. Let p, q E A + n A ~ (Fig. 1 ), and let

Pl = @(Ph 41= Q(4).

Since D is a simply connected open set with zr ED, Q(D) is also a simply connected open set containing z,. Because of (28), we have

It follows that if p, does not belong to A +, then Q(D) is not a simply connected open set (Fig. 2). Hence, we have

In a similar way, we get

Then we have

@(n+)cn+,

which implies the existence of a minimum set of CD in A +. Hence, we have proved the following

LEMMA 9. Under the conditions mentioned above, there exists a minimum set A4 of @ in FT.


IX. PROPERTY A;

Assume that Lemma 6 or 7 holds, and let K;’ and K;- be defined as in Section VIII.

LEMMA 10. If K,+ and K, possess Property A:, then all solutions of the Duffing equation (1) are bounded for t E R.

Proof: It suffices to prove that for any given constant a>0 there is a constant b > a such that

B,cK; (B, = K,T 1.

Assume the contrary. Then there is a constant r0 > IzO 1 such that

Let

W\K,t 1 n 4, Z 0 for r > rO.

B,\K: = 6 E;., i= 1

where E,fi is a simply connected open set (i= 1, 2,...). Recalling the proof of Lemma 4, we get at least one set, say E,?, for definiteness, such that

a-% n S, f 0, a$ n S, # 0.

Since K,+ possesses Property At, then for any constant r, > r0 there is a fixed point z, E E,?,, such that

aE,tl, n S,, = 0.

It follows from Lemma 9 that there is a minimum set M, c R[r,, rz] of @ such that

provided that r3 is large enough. Note that F&, is simply connected, and

Ml = KG n RCr,, r21.

Since F;,i, c aKA and KG n aE;, , c aKr: are simply connected closed sets and aK,: is connected with

(K; naE,:,,)nS,Z0,

202 TONGREN DING

there is a connected closed set F, in c?K; n R[v,, r3], which contains M, and intersects S,,. Moreover, F, can be chosen to be simply connected since F,:,iI is simply connected.

Similarly, using Lemma 9, we get a minimum set M, c KA n R[d,, d,] for 0 and a connected closed set F2 in aK; n R[r, , d, 1, which contains M, and intersects S,,.

Without loss of generality, assume

rO<r, <r,<r,<d,<d, cd,.

such that (11) and (14) are satisfied. Then,

Since

there is a continuous and simple curve C in (EL, u S,) n R[r,, d,] connecting S,, and S,.

Then, using a similar technique employed in the proof of Lemma 4 and replacing the fixed points z, and z2 by the minimum sets M, and M,, we get a contradiction. Lemma 10 is then proved.

X. MAIN THEOREMS

Now we are in a position to prove the following main theorems of this paper.

THEOREM 1. Zf p(t) is even, then all solutions of the Duffing equation (1) are bounded for t E R.

Proof: It follows from Lemma 6 that K;t and K, are mutually symmetric with respect to the x axis. Hence, both K,+ and K; satisfy Property A, or Property A:, or Property A$. Then Theorem 1 is an immediate con- sequence of Lemmas 4, 5, and 10.

Similarly, we can prove

THEOREM 2. Zfp(t) is odd and g(x) is odd, then all solutions of the Duff- ing equation (1) are bounded for t E R.


APPENDIX

In this Appendix, we prove

PROPOSITION 3. If E c R2 is a bounded, connected open set and satisfies

En U-G) f 0 fork=1,2,

then there is a connected closed set A c aEn iV with A n L(x,) # fzr (k = 1,2), where L(xk) and Ware defined in Section II.

Proof: Recall that a set Fc R2 is connected if and only if it is impossible to find two disjoint open sets G, and G, whose union contains F while neither G, nor G2 alone contains F.

Let F be any set in IR’, and let x, y E F. If it is impossible to find two disjoint open sets G, and G, such that

XEG,, ~~(325 and Fc G, v G2,

then x is said to be connected with y in F, and this relation is denoted by the following simple notation

x : y.

Then it follows that

(!{ ;; >;;;nyyx_‘:xI

(iii) if x m Fy and y w F~, then x N F~.

This equivalence relationship induces the following partition of F:

where A is some index set with

(1) F,nFg=O whenever a#fi and a,fl~A; (2) x,y~f;, if and only if x wFy.

(see C61).

LEMMA A. If F is closed, then F, is closed for any c1 E A.

In fact, if F, is not closed for some a E A, then there is a sequence

204 TONGREN DING

{x,}cFsuchthatx,+x,asn-+co,andx,$F,.Since (xk}CF,CFand F is closed, we have

XOEFO for some b E d, with /3 # a.

Hence x1 is not connected with x0 in F. There are two disjoint sets G, and Go such that

x,sG,, xo~Go, and PC G, u Go.

Since x,EF~ (n> 1) or x, mFx, (n> l), we have

x,~Gl and x,$Go.

On the other hand, since x, + x0 and x0 E Go, there is a positive integer no such that

x,~Go whenever IZ > no.

This contradiction proves Lemma A.

LEMMA B. if F is closed and bounded, then Fn is connected for any c1 E A.

Assume on the contrary that FE is not connected for some a E A. Then there are two disjoint open sets G, and G2 such that

G,nF,f0, G2nFaf0, and F,cGluG2.

Note that F, is bounded and closed. Then we can assume that the above open sets G, and G2 are bounded. Hence, we have

dist(F,, 8G,) > 0, dist(F,, aG,) > 0.

Then we can assume further that

G,nG,=@.

It follows that, for any point p E F, we can choose a neighborhood N, of p satisfying

N,nG,=@ or N,nG,=@.

Set

UN, 1 N,n(G,uG,)=@ ’

DUFFING’S EQUATlON 205

Then G: and G: are two disjoint open sets whose union contains F while

FnGfxF,nG#@, F~G:IF,~G,#@.

Let XEF~~G, and yEF,nG,. Then XEG: and LEG;. It follows that x is not connected with y in F. This contradicts that x, y E F, if and only if x is connected with y in F. Lemma B is thus proved.

Now, we are going to prove the above-mentioned Proposition 3. Let E, 1 E be a simply connected open set with i3E, c aE, and let

Then F is bounded and closed. From Lemma B, we have

where F, is connected and compact for any u E A. Since E, is simply connected, the relation

F,c W

does not hold for any c1 E A. Then, for any a E A, we have

F,nL(x,)#0 or F, n WJ Z 0.

We claim that there is at least one CI E A satisfying

F,nL(x,)#0 and F, n Wd f 0.

On the contrary, we have the following alternatives:

(I) F,nL(x,)#@ and F,nL(x,)=@, or (II) F, n L(x2) = 0 and F%n L(xz) # 0, for any a E A. Hence, we

have

where

F=H,vH,,

H, = UK, (in which F,, are in case (I))

and

Hz = UK, (in which F,, are in case (II)).

Then H, and H, are disjoint.

206 TONGREN DING

We want to prove that H, and H, are closed. Assume H, is not closed. Then there is a sequence {x,} c Hi with

x, -+ x,, as n + cc such that x,, 4 H,. Since F is closed, we have

X~E H,.

Then there is an index G$ in case (II) such that

xo E Q

Let x, E F,;, where Fx; is in case (I) (n = 1, 2,...), and let

a, E Ux,)nF,; (n = 1, 2,...).

Then {a,} is bounded. Without loss of generality, we can assume

4, + a0 asn-+co.

Since ,5(x,) n F is closed, a, E FE: for some LX: in case (I). It follows that a, is not connected with x0 in F. Then there are two dis-

joint open sets Gy and Gi with

aoEG?, XOE G;, FcGyuG;.

Since a,, + a, and x, + x0 as n + co, there is a positive integer no such that

a,,EGY, x,EG; whenever n > no,

which together with the properties of GT and Gt implies that a, is not connected with X, in F. This is, however, in conflict with

a,, x,EF,;.

Hence, H, is closed. Similarly, it can be proved that H, is closed. Hence, H, and H, are two disjoint compact sets with the properties that

HI UWI) and Hz u Uxz)

are connected sets. Therefore, we can construct two open sets (neighborhoods) I’, and V, such that

V,xHH,, V,lHH,, V,nV,=@, V,nL(x,)=@, V,nL(x,)=@.

Set

G,=V,u{(x,y)lx<x,, -~<y<~},

G2= V,u {(x,y)lx>x,, -cc <~<a}.


Then G, and G, are two disjoint open sets whose union contains aE, . This proves that dE, is not connected. However, this conclusion contradicts that E, is simply connected. Hence, there is at least one F, such that F, is a connected closed set in iYE, n F with

FznUxl)fO, F, n UxJ Z 0.

Since aE, c i?E, then F, c i3E n @. Let

A=F,.

Then we arrive at the desired conclusion of Proposition 3.

ACKNOWLEDGMENTS

The author is very grateful to Professor George Sell for his many helpful suggestions and strong encouragement, and also in debt to Professor Shui-nee Chow for valuable discussions about Section VIII of this paper.

REFERENCES

I. J. MOSER, “Stable and Random Motions in Dynamical Systems,” Annals of Mathematics Studies, Princeton Univ. Press, Princeton, N.J., 1973.

2. L. MARKUS, “Lectures in Differentiable Dynamics,” Amer. Math. Sot., Providence, RI., 1969; rev. ed., 1980.

3. J. LITTLEWOOD, “Some Problems in Real and Complex Analysis,” Heath, Lexington, Mass., 1968.

4. T. DING, Stable sets for measure-preserving homeomorphisms with applications to Hamiltonian systems, unpublished.

5. G. MORRIS, Bounds on solutions of forced conservative oscillators, BUN. Austral. Math. Sot. 14 (1976), 71-93.

6. J. KELLEY, “General Topology,” Van Nostrand-Reinhold, New York, 1955. 7. W. DING, Fixed points of twist mappings and periodic solutions of ordinary differential

equations, Acra Math. Sinica 25 (1981), 227-235.

boundedness of solutions of duffing’s equation* · journal of differential equations 61, 178-207...

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